The following augmented chain complex (of chain complexes) is exact (see [1]}):
\(0 \xleftarrow{} C_*(\cup _{j\in J} X^j) \xleftarrow{} C_{0,*} \xleftarrow{} C_{1,*} \xleftarrow{} \cdots \xleftarrow{} C_{p-1,*} \xleftarrow{} C_{p,*} \xleftarrow{} \cdots \) 
